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Problem 1. (a) Describe an algorithm that sorts an input array A[1... n] by calling a subroutine SQRTSORT (k), which sorts the subarray A[k+1...k+ vn] in place, given an arbitrary integer k between 0 and n - Vn as input. (To simplify the problem, assume that Vn is an integer.) Your algorithm is only allowed to inspect or modify the input array by calling SQRISORT; in particular, your algorithm must not directly compare, move or copy array elements. How many times does your algorithm call SQRTSORT in the worst case? Give pseudocode. Remem- ber that you cannot use anything other than calling the SQRTSORT routine and maybe some loops. No formal proof of correctness is mandatory. But write a few sentences justifying your approach. Prove that your algorithm from part (a) is optimal up to constant factors. In (b) other words, if f(n) is the number of times your algorithm calls SQRTSORT, prove that no algorithm can sort using o(f (n)) calls to SQRTSORT. Note that here we are assuming that these algorithms cannot do anything other than calling SQRTSORT repeatedly. (Hint: Think of a typical worst case example for any sorting algorithm) (c) Now suppose SQRTSORT is implemented recursively, by calling your sorting algo- rithm from part (a). For example, at the second level of recursion, the algorithm is sorting arrays roughly of size n/4. What is the worst-case running time of the resulting sorting algorithm? (To simplify the analysis, assume that the array size n has the form 22"), so that repeated square roots are always integers.)

Problem 1. (a) Describe an algorithm that sorts an input array A[1... n] by calling a subroutine SQRTSORT (k), which sorts the subarray A[k+1...k+ vn] in place, given an arbitrary integer k between 0 and n - Vn as input. (To simplify the problem, assume that Vn is an integer.) Your algorithm is only allowed to inspect or modify the input array by calling SQRISORT; in particular, your algorithm must not directly compare, move or copy array elements. How many times does your algorithm call SQRTSORT in the worst case? Give pseudocode. Remem- ber that you cannot use anything other than calling the SQRTSORT routine and maybe some loops. No formal proof of correctness is mandatory. But write a few sentences justifying your approach. Prove that your algorithm from part (a) is optimal up to constant factors. In (b) other words, if f(n) is the number of times your algorithm calls SQRTSORT, prove that no algorithm can sort using o(f (n)) calls to SQRTSORT. Note that here we are assuming that these algorithms cannot do anything other than calling SQRTSORT repeatedly. (Hint: Think of a typical worst case example for any sorting algorithm) (c) Now suppose SQRTSORT is implemented recursively, by calling your sorting algo- rithm from part (a). For example, at the second level of recursion, the algorithm is sorting arrays roughly of size n/4. What is the worst-case running time of the resulting sorting algorithm? (To simplify the analysis, assume that the array size n has the form 22"), so that repeated square roots are always integers.)

Computer Networking: A Top-Down Approach (7th Edition)
Computer Networking: A Top-Down Approach (7th Edition)
7th Edition
ISBN:9780133594140
Author:James Kurose, Keith Ross
Publisher:James Kurose, Keith Ross
Chapter1: Computer Networks And The Internet
Section: Chapter Questions
Problem R1RQ: What is the difference between a host and an end system? List several different types of end...
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Problem 1. Describe an algorithm that sorts an input array A[1 · . · n] by calling a subroutine (a) SQRTSORT (k), which sorts the subarray A[k+1...k+ vn] in place, given an arbitrary integer k between 0 and n Vn as input. (To simplify the problem, assume that n is an integer.) Your algorithm is only allowed to inspect or modify the input array by calling SQRTSORT; in particular, your algorithm must not directly compare, move or copy array elements. How many times does your algorithm call SQRTSORT in the worst case? Give pseudocode. Remem- ber that you cannot use anything other than calling the SQRTSORT routine and maybe some loops. No formal proof of correctness is mandatory. But write a few sentences justifying your approach. Prove that your algorithm from part (a) is optimal up to constant factors. In (b) other words, if f(n) is the number of times your algorithm calls SQRTSORT, prove that no algorithm can sort using o(f(n)) calls to SQRTSORT. Note that here we are assum these algorithms cannot do anything other than calling SQRTSORT repeatedly. that (Hint: Think of a typical worst case example for any sorting algorithm) Now suppose SQRTSORT is implemented recursively, by calling your sorting algo- (c) rithm from part (a). For example, at the second level of recursion, the algorithm is sorting arrays roughly of size n'/4. What is the worst-case running time of the resulting sorting algorithm? (To simplify the analysis, assume that the array size n has the form 22"), so that repeated square roots are always integers.)
Transcribed Image Text:Problem 1. Describe an algorithm that sorts an input array A[1 · . · n] by calling a subroutine (a) SQRTSORT (k), which sorts the subarray A[k+1...k+ vn] in place, given an arbitrary integer k between 0 and n Vn as input. (To simplify the problem, assume that n is an integer.) Your algorithm is only allowed to inspect or modify the input array by calling SQRTSORT; in particular, your algorithm must not directly compare, move or copy array elements. How many times does your algorithm call SQRTSORT in the worst case? Give pseudocode. Remem- ber that you cannot use anything other than calling the SQRTSORT routine and maybe some loops. No formal proof of correctness is mandatory. But write a few sentences justifying your approach. Prove that your algorithm from part (a) is optimal up to constant factors. In (b) other words, if f(n) is the number of times your algorithm calls SQRTSORT, prove that no algorithm can sort using o(f(n)) calls to SQRTSORT. Note that here we are assum these algorithms cannot do anything other than calling SQRTSORT repeatedly. that (Hint: Think of a typical worst case example for any sorting algorithm) Now suppose SQRTSORT is implemented recursively, by calling your sorting algo- (c) rithm from part (a). For example, at the second level of recursion, the algorithm is sorting arrays roughly of size n'/4. What is the worst-case running time of the resulting sorting algorithm? (To simplify the analysis, assume that the array size n has the form 22"), so that repeated square roots are always integers.)
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